Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry. The method of moving frames, a natural means for discovering and proving important results, provides the basis of treatment for topics discussed. Its application in many areas helps to connect the various geometries and to uncover many deep relationships, such as the Lawson correspondence. The nearly 300 problems and exercises range from simple applications to open problems. Exercises are embedded in the text as essential parts of the exposition. Problems are collected at the end of each chapter; solutions to select problems are given at the end of the book. Mathematica (R), Matlab (TM), and Xfig are used to illustrate selected concepts and results. The careful selection of results serves to show the reader how to prove the most important theorems in the subject, which may become the foundation of future progress. The book pursues significant results beyond the standard topics of an introductory differential geometry course. A sample of these results includes the Willmore functional, the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature immersions, isothermic immersions, and the duality between minimal surfaces in Euclidean space and constant mean curvature surfaces in hyperbolic space. The book concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to emphasize the geometric interpretation of invariants and other constructions. Working in low dimensions helps students develop a strong geometric intuition. Aspiring geometers will acquire a working knowledge of curves and surfaces in classical geometries. Students will learn the invariants of conformal geometry and how these relate to the invariants of Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie sphere geometry, which require the notion of Legendre immersions of a contact structure. Prerequisites include a completed one semester standard course on manifold theory.

Excellent teaching of mathematics at the elementary school level requires teachers to be experts in school mathematics. This textbook helps prospective teachers achieve the necessary expertise by presenting topics from the K-6 mathematics curriculum at a greater depth than is found in most classrooms. The knowledge that comes from this approach gives prospective teachers essential insight into how topics interrelate and where difficulties may lie. Information is presented at a pace that makes it interesting, rewarding, and enjoyable.With the deeper mathematical preparation inherent in this book, prospective teachers will come away knowing how to explain concepts, demonstrate computational procedures, and lead students through problem-solving techniques. Both students and teachers will find this book key to learning the necessary material and knowing how to express it at the right level. The primary focus is on the foundations of arithmetic, along with a selection of topics from geometry, and a wide range of applications. The number line is used throughout to visualize concepts and to tie them to solutions. The book emphasizes explanations of concepts, of how to solve problems, and of how the concepts relate to the solutions of the problems. This is a textbook for a college course in mathematics for prospective elementary school teachers. It will also be an excellent reference source for instructors of such courses.

The theory of exterior differential systems provides a framework for systematically addressing the typically non-linear, and frequently overdetermined, partial differential equations that arise in differential geometry. Adaptation of the techniques of microlocalization to differential systems have led to recent activity on the foundations of the theory; in particular, the fundamental role of the characteristic variety in geometric problems is now clearly established. In this book the general theory is explained in a relatively quick and concrete manner, and then this general theory is applied to the recent developments in the classical problem of isometric embeddings of Riemannian manifolds.